{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T00:41:29Z","timestamp":1768610489472,"version":"3.49.0"},"reference-count":43,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T00:00:00Z","timestamp":1663891200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of the Hunan Province","award":["11972379"],"award-info":[{"award-number":["11972379"]}]},{"name":"Natural Science Foundation of the Hunan Province","award":["2020SK2060"],"award-info":[{"award-number":["2020SK2060"]}]},{"name":"Natural Science Foundation of the Hunan Province","award":["2021JJ10061"],"award-info":[{"award-number":["2021JJ10061"]}]},{"name":"Natural Science Foundation of the Hunan Province","award":["502390001"],"award-info":[{"award-number":["502390001"]}]},{"name":"Key R&amp;D Program of the Hunan Province","award":["11972379"],"award-info":[{"award-number":["11972379"]}]},{"name":"Key R&amp;D Program of the Hunan Province","award":["2020SK2060"],"award-info":[{"award-number":["2020SK2060"]}]},{"name":"Key R&amp;D Program of the Hunan Province","award":["2021JJ10061"],"award-info":[{"award-number":["2021JJ10061"]}]},{"name":"Key R&amp;D Program of the Hunan Province","award":["502390001"],"award-info":[{"award-number":["502390001"]}]},{"name":"Hunan Science Fund for Distinguished Young Scholars","award":["11972379"],"award-info":[{"award-number":["11972379"]}]},{"name":"Hunan Science Fund for Distinguished Young Scholars","award":["2020SK2060"],"award-info":[{"award-number":["2020SK2060"]}]},{"name":"Hunan Science Fund for Distinguished Young Scholars","award":["2021JJ10061"],"award-info":[{"award-number":["2021JJ10061"]}]},{"name":"Hunan Science Fund for Distinguished Young Scholars","award":["502390001"],"award-info":[{"award-number":["502390001"]}]},{"name":"Central South University Research","award":["11972379"],"award-info":[{"award-number":["11972379"]}]},{"name":"Central South University Research","award":["2020SK2060"],"award-info":[{"award-number":["2020SK2060"]}]},{"name":"Central South University Research","award":["2021JJ10061"],"award-info":[{"award-number":["2021JJ10061"]}]},{"name":"Central South University Research","award":["502390001"],"award-info":[{"award-number":["502390001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>As a type of urban life project in China, bridges need a certain capacity of trains running safely after an earthquake to ensure and guarantee transportation on railway lines, post-disaster reconstruction and relief work. Since aftershocks may occur after the main shock, the earthquake-induced irregularity and aftershock intensity are fully considered, based on the running safety index in the seismic design of bridges. However, there is a lack of research on the running safety of trains after an earthquake; it is mainly judged on experience, and lacks theoretical basis. In this paper, the established finite element model of a train bridge interaction system with symmetry was considered. The point estimation method (PEM) combined with moment expansion approximation (MEA) is used for random calculation of the Housner Intensity (HI). Furthermore, running safety indexes were analyzed and the running safety performance of a simply supported bridge with symmetry was assessed under a post-earthquake condition. Then the limit value, to ensure the traffic safety performance after an earthquake, is calculated based on stochastic analysis. The HI can be calculated with full consideration of the randomness of aftershock intensity and structural parameters. On this basis, a calculation method of the HI that considers the randomness of aftershock intensity is proposed. This study can be helpful for the performance-based design of symmetric railway structures under post-earthquake conditions.<\/jats:p>","DOI":"10.3390\/sym14101998","type":"journal-article","created":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T01:23:16Z","timestamp":1664414596000},"page":"1998","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Stochastic Analysis of Train Running Safety on Bridge with Earthquake-Induced Irregularity under Aftershock"],"prefix":"10.3390","volume":"14","author":[{"given":"Jincheng","family":"Tan","sequence":"first","affiliation":[{"name":"School of Civil Engineering, Central South University, Changsha 410075, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1636-4111","authenticated-orcid":false,"given":"Ping","family":"Xiang","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Central South University, Changsha 410075, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0355-643X","authenticated-orcid":false,"given":"Han","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Central South University, Changsha 410075, China"}]},{"given":"Jian","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Central South University, Changsha 410075, China"}]},{"given":"Bailong","family":"Ye","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Central South University, Changsha 410075, China"}]},{"given":"Delei","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Building Engineering, University of Huanghuai, No. 6 Kaiyuan Road, Zhumadian 463000, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,23]]},"reference":[{"key":"ref_1","unstructured":"Mu, X. (2009, January 5). On disaster prevention and mitigation of urban Bridges based on lifeline engineering. Proceedings of the Special for the Fourth National (International) Technical Summit Conference & the West Communication Scientific Technical Innovation Conference, Chongqing, China."},{"key":"ref_2","unstructured":"Dai, S. (2008, January 27\u201328). Thoughts on earthquake rescue and communication of Qingjiang No. 7 bridge on Baocheng railway. Proceedings of the Symposium on the Impact of Earthquake Disasters on Railways and Countermeasures, Chengdu, China."},{"key":"ref_3","first-page":"113","article-title":"Seismic damage evaluation of high-speed railway bridge components under different intensities of earthquake excitations","volume":"152","author":"Kanga","year":"2017","journal-title":"Eng. Struct."},{"key":"ref_4","first-page":"1","article-title":"Earthquake Influence on the Rail Irregularity on High-Speed Railway Bridge","volume":"2020","author":"Lai","year":"2020","journal-title":"Shock. Vib."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Jiang, L.Z., Yu, J., Zhou, W.B., Yan, W.J., Lai, Z.P., and Feng, Y.L. (2020). Applicability analysis of high-speed railway system under the action of near-fault ground motion. Soil Dyn. Earthq. Eng., 139.","DOI":"10.1016\/j.soildyn.2020.106289"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3215","DOI":"10.1080\/00423114.2021.1942507","article-title":"Lateral girder displacement effect on the safety and comfortability of the high-speed rail train operation","volume":"60","author":"Lai","year":"2021","journal-title":"Veh. Syst. Dyn."},{"key":"ref_7","first-page":"1","article-title":"Safety and comfort assessment of a train passing over an earthquake-damaged bridge based on a probability model","volume":"17","author":"Liu","year":"2021","journal-title":"Struct. Infrastruct. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Yu, J., Jiang, L., and Zhou, W. (2022). Study of the Target Earthquake-Induced Track Irregularity Spectrum under Transverse Random Earthquakes. Int. J. Struct. Stab. Dyn., 22.","DOI":"10.1142\/S0219455422501905"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Yu, J., Jiang, L., Zhou, W., Liu, X., and Lai, Z. (2022). Seismic-Induced Geometric Irregularity of Rail Alignment under Transverse Random Earthquake. J. Earthq. Eng., 1\u201322.","DOI":"10.1080\/13632469.2022.2030437"},{"key":"ref_10","unstructured":"Institute, R.T.R. (2012). Design Standards for Railway Structures and Commentary (Seismic Design), Maruzen."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/j.soildyn.2004.10.005","article-title":"Study on methodology for running safety assessment of trains in seismic design of railway structures","volume":"25","author":"Luo","year":"2005","journal-title":"Soil Dyn. Earthq. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Zhao, H., Wei, B., Jiang, L., and Xiang, P. (2022). Seismic running safety assessment for stochastic vibration of train\u2013bridge coupled system. Arch. Civ. Mech. Eng., 22.","DOI":"10.1007\/s43452-022-00451-3"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2311","DOI":"10.1002\/eqe.2188","article-title":"Effects of aftershocks on peak ductility demand due to strong ground motion records from shallow crustal earthquakes","volume":"41","author":"Goda","year":"2012","journal-title":"Earthq. Eng. Struct. Dyn."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.probengmech.2016.08.003","article-title":"Random dynamic analysis of a train-bridge coupled system involving random system parameters based on probability density evolution method","volume":"46","author":"Mao","year":"2016","journal-title":"Probabilistic Eng. Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.apm.2018.01.038","article-title":"A stochastic dynamic model of train-track-bridge coupled system based on probability density evolution method","volume":"59","author":"Yu","year":"2018","journal-title":"Appl. Math. Model."},{"key":"ref_16","first-page":"1387","article-title":"Probability Density Evolution of Stochastic Structural Responses","volume":"31","author":"Jie","year":"2003","journal-title":"J. Tongji Univ."},{"key":"ref_17","first-page":"1","article-title":"Application of KLE-PEM for Random Dynamic Analysis of Nonlinear Train-Track-Bridge System","volume":"2020","author":"Jiang","year":"2020","journal-title":"Shock. Vib."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.jsv.2017.07.016","article-title":"Dimension reduction of Karhunen-Loeve expansion for simulation of stochastic processes","volume":"408","author":"Liu","year":"2017","journal-title":"J. Sound Vib."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Jiang, L.Z., Liu, X., Xiang, P., and Zhou, W.B. (2019). Train-bridge system dynamics analysis with uncertain parameters based on new point estimate method. Eng. Struct., 199.","DOI":"10.1016\/j.engstruct.2019.109454"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Liu, X., Jiang, L.Z., Lai, Z.P., Xiang, P., and Chen, Y.J. (2020). Sensitivity and dynamic analysis of train-bridge coupled system with multiple random factors. Eng. Struct., 221.","DOI":"10.1016\/j.engstruct.2020.111083"},{"key":"ref_21","first-page":"31","article-title":"A stochastic finite element method for dynamic analysis of bridge structures under moving loads","volume":"82","author":"Liu","year":"2022","journal-title":"Struct. Eng. Mech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.apm.2017.06.027","article-title":"A probabilistic model for track random irregularities in vehicle\/track coupled dynamics","volume":"51","author":"Xu","year":"2017","journal-title":"Appl. Math. Model."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Xu, L., Zhao, Y., Zhu, Z., Li, Z., Liu, H., and Yu, Z. (2021). Vehicle-track random vibrations considering spatial frequency coherence of track irregularitives. Veh. Syst. Dyn., 1\u201322.","DOI":"10.1080\/00423114.2021.1986224"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Xiang, P., Huang, W., Jiang, L., Lu, D., Liu, X., and Zhang, Q. (2021). Investigations on the influence of prestressed concrete creep on train-track-bridge system. Constr. Build. Mater., 293.","DOI":"10.1016\/j.conbuildmat.2021.123504"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2014\/304219","article-title":"Numerical Simulation of Vertical Random Vibration of Train-Slab Track-Bridge Interaction System by PEM","volume":"2014","author":"Zeng","year":"2014","journal-title":"Shock. Vib."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Liu, X., Xiang, P., Jiang, L.Z., Lai, Z.P., Zhou, T., and Chen, Y.J. (2020). Stochastic Analysis of Train-Bridge System Using the Karhunen-Loeve Expansion and the Point Estimate Method. Int. J. Struct. Stab. Dyn., 20.","DOI":"10.1142\/S021945542050025X"},{"key":"ref_27","unstructured":"Burden, R.L., Faires, J.D., and Burden, A.M. (2015). Numerical Analysis, Cengage Learning."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Wang, H.-P., Chen, H., Chen, C., Zhang, H.-Y., Jiang, H., Song, T., and Feng, S.-Y. (2021). The Structural Performance of CFRP Composite Plates Assembled with Fiber Bragg Grating Sensors. Symmetry, 13.","DOI":"10.3390\/sym13091631"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3624","DOI":"10.1109\/JLT.2018.2838122","article-title":"Optical Fiber Sensor Based In-Field Structural Performance Monitoring of Multilayered Asphalt Pavement","volume":"36","author":"Wang","year":"2018","journal-title":"J. Lightwave Technol."},{"key":"ref_30","unstructured":"Lizhong, J., Jian, Y., and Wangbao, Z. (2022). Study on geometrical irregularity of rail induced by transverse earthquake. Eng. Mech., 39."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3779","DOI":"10.1007\/s10518-021-01125-w","article-title":"Running test on high-speed railway track-simply supported girder bridge systems under seismic action","volume":"19","author":"Yu","year":"2021","journal-title":"Bull. Earthq. Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.probengmech.2015.10.005","article-title":"Adaptive estimation of statistical moments of the responses of random systems","volume":"43","author":"Fan","year":"2016","journal-title":"Probabilistic Eng. Mech."},{"key":"ref_33","unstructured":"Castillo, E. (2012). Extreme Value Theory in Engineering, Elsevier."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1061\/(ASCE)0733-9399(2000)126:4(433)","article-title":"New point estimates for probability moments","volume":"126","author":"Zhao","year":"2000","journal-title":"J. Eng. Mech."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2016\/2831206","article-title":"Theory and implementation of a two-step unconditionally stable explicit integration algorithm for vibration analysis of structures","volume":"2016","author":"Changqing","year":"2016","journal-title":"Shock. Vib."},{"key":"ref_36","unstructured":"Zeng, Q., and Guo, X. (1999). Theory and Application of Vibration Analysis for Train Bridge Time-History System, China Railway Publishing."},{"key":"ref_37","unstructured":"Zhai, W.M., and Xia, H. (2011). Train-Track-Bridge Dynamic Interaction: Theory and Engineering Application, Science Press."},{"key":"ref_38","first-page":"011001.1","article-title":"Development of Vehicle Dynamics Simulation for Safety Analyses of Rail Vehicles on Excited Tracks","volume":"4","author":"Nishimura","year":"2009","journal-title":"J. Comput. Nonlinear Dyn."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.12989\/sem.2015.54.5.1017","article-title":"Random vibration analysis of train-slab track-bridge coupling system under earthquakes","volume":"54","author":"Zeng","year":"2015","journal-title":"Struct. Eng. Mech."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1016\/0045-7949(94)90422-7","article-title":"Structural responses to arbitrarily coherent stationary random excitations","volume":"50","author":"Lin","year":"1994","journal-title":"Comput. Struct."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"129","DOI":"10.2219\/rtriqr.48.129","article-title":"Method for running safety assessment of railway vehicles against structural vibration displacement during earthquakes","volume":"48","author":"Luo","year":"2007","journal-title":"Q. Rep. RTRI"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"968","DOI":"10.1007\/s11771-021-4657-2","article-title":"Dynamic response limit of high-speed railway bridge under earthquake considering running safety performance of train","volume":"28","author":"Liu","year":"2021","journal-title":"J. Cent. South Univ."},{"key":"ref_43","first-page":"185","article-title":"Correlation between ground motion intensity indices and sdof system responses with medium-to-long period based on the wenchuan earthquake data","volume":"28","author":"Han","year":"2011","journal-title":"Eng. Mech."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/1998\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:38:25Z","timestamp":1760143105000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/1998"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,23]]},"references-count":43,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14101998"],"URL":"https:\/\/doi.org\/10.3390\/sym14101998","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,23]]}}}